Problem: Select all polynomials that are divisible by $(x+1)$. Choose all answers that apply: Choose all answers that apply: (Choice A) A $A(x)=x^4-3x^3-4$ (Choice B) B $B(x)=x^4+2x+1$ (Choice C) C $C(x)=x^3-3x^2+4x-2$ (Choice D) D $D(x)=x^3+2x^2-5x-6$
Solution: The following statements are equivalent: $(x+1)$ is a factor of $p(x)$ $p(x)$ is divisible by $(x+1)$ The remainder of $\dfrac{p(x)}{x+1}$ is $0$ We can use the polynomial remainder theorem to solve this problem: For a polynomial $p(x)$ and a number $a$, the remainder on division by $x-a$ is $p(a)$. According to the theorem, the remainder when $p(x)$ is divided by $(x+1)$, which can be rewritten as $(x-({-1}))$, is equal to $p({-1})$. So to check each polynomial is divisible by $(x+1)$, we need to check if that polynomial's value at ${x=-1}$ is zero. $\begin{aligned} A({-1})&=0 \\\\ B({-1})&=0 \\\\ C({-1})&=-10 \\\\ D({-1})&=0 \end{aligned}$ In conclusion, the following polynomials are divisible by $(x+1)$ : $A(x)=x^4-3x^3-4$ $B(x)=x^4+2x+1$ $D(x)=x^3+2x^2-5x-6$